VISUALISING FRACTIONS OPERATIONS

Misconceptions in mathematics are common, and are usually misunderstanding and/or misinterpretation based on incorrect meanings. Knowing the nature of a misconception and its source would help to fathom ways of planning appropriate instruction that is beneficial to students. A likely misconception to see in my math class is   This misconception could be associated with students transitioning from operations with whole numbers to operations with fractions because, to them, the rules have changed. Because of previous “knowledge”, the students performed operations distinctly, considering the numerators and the denominators in separate operation. In order to teach additions and subtraction involving fractions, it is important to impress upon the students that the numerator indicates the number of parts and the denominator indicates the types of part (the whole). The computational rule usually don’t help students think about the operations and what they mean. When the students are only armed with rules, even when the rules are mastered, it is quickly lost in the short term as the myriad of rules soon become meaningless when mixed together (On complex operations).

I am using the 3-step T-GEM cycle to create activities that cover operations involving fractions.

To generate relationship: the students will begin with simple contextual tasks. The students will challenge themselves with an interactive activity on Phet Interactive Simulations website. The goal is to enable them to find matching fractions using numbers and pictures, to make the same fractions using different numbers, to match fractions in different picture patterns, to compare fractions using numbers and patterns. Here is a link to the activity.

To evaluate the relationship: at this point the students are expected to have a conceptual understanding of fractions that is the numerator indicates the number of parts and the denominator indicates the types of part (the whole). The students will engage in operations involving fractions. Because the students usually handle addition better than subtraction, they will start with addition, adding fraction with same denominator. The vast majority of my students can tell that a half plus a half is one or one whole. I will write down the question and apply their general strategy (the misconception stated earlier), then interesting questions will raised.

This example will shake the misconception. The students will be encouraged to connect the meaning of fraction computation with whole number computation such as ¼ is one part out of four equal parts of a whole. In order to practise addition of fractions with same and different denominators, the students will use the interactive activity “Fraction wall” . The activity illustrates the addition of fractions using array technique. The students will explore similar array techniques to perform subtraction. An illustration of using array technique is

The aim of this model is to help the students learn to think about fraction and the operation, to contribute to mental methods, and provide a useful background when they eventually learn the standard algorithms of LCM.

To modify the relationship: the students will reflect on the work by using an interactive activity that allow them to visualise an array of fractions, compare, add, subtract, and multiply two fractions with animations providing a conceptual understanding over the use of LCM to perform fractions operations.  Here is a link to the activity.

Reference:

Khan, S. (2011). New pedagogies on teaching science with computer simulations. Journal of Science Education and Technology, 20(3), 215-232.